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We study the notion of safe realizability for high-level message sequence charts (HMSCs) (Proceedings of the 28th International Colloquium on Automata, Languages and Programming (ICALP 2001), Crete (Greece), Lecture Notes in Computer Science, Vol. 2076, Springer, Berlin, 2001, pp. 797–808). We show that safe realizability is EXPSPACE-complete for bounded HMSCs but undecidable for the class of all HMSCs. This solves two open problems from Alur et al. Moreover we prove that safe realizability is also EXPSPACE-complete for the larger class of globally-cooperative HMSCs. [Copyright &y& Elsevier]